tan ^ 1 #160; ( 3 x 6 x^ 2 5x 3 #160; ) #160; compare it with , tan ^ 1 #160; ( a b 1+ab #160; ) #160; ⇒ 1+ab=6 x^ ...

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Basic Inverse Trigonometric Functions

tan-15-x6x2-5...

Question

${tan}^{- 1} (\frac{5 - x}{6 x^{2} - 5 x - 3})$ .

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Solution

$\begin{matrix} {tan}^{- 1} (\frac{3 - x}{6 x^{2} - 5 x - 3}) c o m p a r e i t w i t h, {tan}^{- 1} (\frac{a - b}{1 + a b}) \Rightarrow 1 + a b = 6 x^{2} - 5 x - 3 \Rightarrow 6 x^{2} - 5 x - 3 = 1 + a b \Rightarrow 6 x^{2} - 5 x - 4 = a b \Rightarrow (3 x - 4) (2 x + 1) = a b A l s o 5 - x = 2 x + 1 - (3 x - 4) i . e, a = 2 x + 1 b = 3 x - 4 {tan}^{- 1} (\frac{a - b}{1 + a b}) = {tan}^{- 1} a + {tan}^{- 1} b \Rightarrow {tan}^{- 1} (2 x + 1) + {tan}^{- 1} (3 x - 4) \end{matrix}$

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